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Lecture 1. bearing capacity of soil
1. University of Warith Al Anbiya’a
College of Engineering
Civil Engineering Department
4th Year Class
Foundation Engineering I
2. Introduction
Foundations are structural elements, which are designed to transfer building loads safely
to the soil. They must satisfy the following two design criteria:
1. Bearing capacity : There must be no shear failure within the soil.
2. Settlement : The settlement must be within tolerable limits.
Foundations are divided in to two types based on depth of embedded to width
ratio Τ
𝑫
𝑩 as:
1. Shallow Foundations:
𝑫
𝑩
=< 𝟒 such as spread ,strip,
continuous ,combined and raft (mat) foundation.
2. Deep Foundations:
𝑫
𝑩
> 𝟒 such as piles and drilled shifts.
Shallow Foundations
Building
Building
Deep Foundations
3. Spread footing Combined footing Strip footing
Strap footing Mat (Raft) foundation
Shallow Foundations:
𝑫
𝑩
=< 𝟒
5. Bearing Capacity of soil :-
(c) Ultimate Bearing Capacity (qult ) :Is the maximum bearing capacity of soil at which the soil fails by shear.
(e) Allowable Bearing Capacity (𝒒𝒂𝒍𝒍) : The maximum allowable net load intensity on the soil allowing for
both shear and settlement and it is simply the ultimate bearing capacity divided by the a factor of safety.
𝒒𝒂𝒍𝒍 =
qult
𝑭. 𝑺
(a) Total Overburden Pressure qo: Is the intensity of total overburden pressure due to the weight of both soil and
water at the base level of the foundation.
(b) Effective Overburden Pressure 𝒒𝒐
′ : Is the intensity of effective overburden pressure due to the weight of
soil only at the base level of the foundation.
(d) The Net Ultimate Bearing Capacity (qnu): Is the bearing capacity in excess of the effective overburden
pressure, expressed:
(f) Net Allowable Bearing Pressure, 𝒒𝒏𝒂 is expressed as
(g) Safe Bearing Pressure, 𝒒𝒔 :is defined as the net safe bearing pressure which produces a settlement of the
foundation which does not exceed a permissible limit.
(h) Gross Bearing Capacity (𝒒𝐠𝐫𝐨𝐬𝐬 ): It is the total unit pressure at the base of footing which the soil can take
up.
Note: In the design of foundations, one has to use the least of the two values of 𝒒𝒏𝒂 and 𝒒𝒔.
qnu=qult - 𝒒𝒐
′
𝒒𝒏𝒂 =
qult
𝑭. 𝑺
=
qult − 𝒒𝒐
′
𝑭. 𝑺
𝒒𝒐
′ = 𝜸′𝑫𝒇
qo= 𝜸𝑫𝒇
6. 2. Local Shear Failure :
Occurs over medium to dense
cohesion less soils and
medium to stiff cohesive soil.
3. Punching Shear Failure :
Occurs loose to very loose
cohesion less soils and soft to
very soft cohesive soil.
1. General Shear Failure :
Occurs over dense to very
dense cohesion less soils and
very stiff to hard cohesive soil.
Types of Shear Failure: Shear Failure: Also called “Bearing capacity failure” and it’s
occur when the shear stresses in the soil exceed the shear
strength of the soil.
7. Determination of the Ultimate Bearing Capacity of Soil:
1. Terzaghi’s Bearing Capacity Theory:Terzaghi was the first to present a comprehensive
theory for evaluation of the ultimate bearing capacity of rough shallow foundation. This
theory is based on the following assumptions:
1. Soil beneath the foundation is a homogeneous , isotropic and infinite half – space.
2. Depth of embedment not more than width (𝑫𝒇 ≤ 𝑩).
3. The load is concentric and vertical.
4. Foundation has a horizontal base on a level ground surface.
5. The failure mode is General Shear Failure .
6. The foundation is strip. ( Τ
𝑩
𝑳 > 𝟓).
qu=𝒄′Nc+𝒒′Nq+0.5B𝜸′Nγ Terzaghi’s Bearing Capacity Equations for strip footing
qu=Ultimate bearing capacity of the 𝐮𝐧𝐝𝐞𝐫𝐥𝐲𝐢𝐧𝐠 soil (KN/𝒎2)
𝒄′=Cohesion of Soil beneath foundation (KN/𝒎2)
𝒒′=𝐄𝐟𝐞𝐞𝐜𝐭𝐢𝐯𝐞 stress at the bottom of the foundation = 𝜸′𝑫𝒇 (KN/𝒎2)
B: least width (or diameter) of footing.
𝜸′:Unit weight of the soil
Nc,Nq,Nγ=Bearing capacity factors (nondimensional)and are functions 𝐨𝐧𝐥𝐲 of the
𝐮𝐧𝐝𝐞𝐫𝐥𝐲𝐢𝐧𝐠 soil friction angle,ϕ,→→ The variations of bearing capacity factors
and underlying soil friction angle are given in (Table.1) for general shear failure.
9. Effect of the footing shape for Terzaghi equation
Footing
Shape
ൗ
𝑩
𝑳
𝑺𝒄 𝑺𝒒 𝑺𝜸 Bearing Capacity Equations
Strip 0 1 1 1 qu=𝒄′Nc+𝒒′Nq+0.5B𝜸′Nγ
Circular -- 1.3 1 0.6 qu=1.3𝒄′Nc+𝒒′Nq+0.3D𝜸′Nγ
Square 1 1.3 1 0.8 qu=𝟏. 𝟑𝒄′Nc+𝒒′Nq+0.4B𝜸′Nγ
Rectangular > 0 𝑎𝑛𝑑 1 < 1+ 0.3
𝐵
𝐿
1 1- 0.2
𝐵
𝐿 qu=(1+ 0.3
𝑩
𝑳
)𝒄′Nc+𝒒′Nq+(1- 0.2
𝑩
𝑳
)0.5B𝜸′Nγ
To estimate the ultimate bearing capacity of square, circular and rectangular foundations,
Terzaghi equation may be respectively modified to qu=𝒄′Nc𝑆𝑐+𝒒′Nq𝑆𝑞+ 0.5B𝜸′Nγ𝑆𝛾
Where 𝑆 𝐢𝐬 𝐬𝐡𝐚𝐩𝐞 𝐟𝐚𝐜𝐭𝐨𝐫
Effect of the type of failure for Terzaghi equation
For local shear failure Terzaghi suggested using of modified cohesion and angle of internal
friction ( ҧ
𝑐 and ഥ
∅ ) as follows:
ҧ
𝑐=
2
3
c 𝑡𝑎𝑛ഥ
∅ =
2
3
𝑡𝑎𝑛∅
10. Effect of Water Table on Bearing Capacity:
Terzaghi and Meyerhof equations give the ultimate bearing capacity based on the assumption
that the water table is located well below the foundation. However, if the water table is close to
the foundation, the bearing capacity will decreases due to the effect of water table, some
modification of the bearing capacity equations (Terzaghi and Meyerhof) will be necessary.
The values which will be modified are:
1. (q for soil above the foundation) in the second term of equations.
2. (γ for the underlying soil) in the third (last) term of equations .
Case I. The water table is located so that 0≤D1≤Df as shown in the following figure:
• For the soil above the foundation) The factor ,q, (second term) q=D1×γd+D2×(γsat−γw)
• For the soil under the foundation) The factor ,γ, (third term) γ′=γsat−γw
Case II. The water table is located so that 0≤d<B as shown in the following figure:
• The factor ,q, (second term)= q=Df×γd
• The factor ,γ, (third term)= ത
γ=γ′+
𝑑
𝐵
(γd − γ′)
Case III. The water table is located so
that d≥B, in this case the
water table is assumed have
no effect on the ultimate
bearing capacity.
11. Net Ultimate Bearing Capacity and Safety Factor
The net ultimate bearing capacity qnu is defined as the pressure at the base level of the
foundation in excess of the effective overburden pressure
The net qnu for a strip footing is
Similar expressions can be written for square, circular, and rectangular foundations and
also for local shear failure conditions.
Allowable Bearing Pressure
The gross allowable bearing pressure is
In the same way the net allowable bearing pressure qna is
where Fs = factor of safety which is normally assumed as equal to 3.
𝒒𝒐
′ = 𝜸′𝑫𝒇
qnu=qult - 𝒒𝒐
′ = qult −𝜸′𝑫𝒇= 𝒄′Nc+𝜸′𝑫𝒇
′
(Nq-1)+0.5B𝜸′Nγ
𝒒𝒂𝒍𝒍 =
qult
𝑭. 𝑺
𝒒𝒏𝒂 =
qult
𝑭. 𝑺
=
qult − 𝒒𝒐
′
𝑭. 𝑺
12. Example (1): Determine the allowable bearing capacity of a strip footing shown below using
Terzaghi Equations if c = 0, φ = 30° , Df = 1.0 m , B = 1.0m , γ soil = 19 kN/𝒎3,
the water table is at ground surface, and SF=3.
qu=𝒄′Nc𝑆𝑐+𝒒′Nq𝑆𝑞+ 0.5B𝜸′Nγ𝑆𝛾
Shape factors: from table , for strip footing 𝑆𝑐=𝑆𝑞= 𝑆𝛾 = 1.0
Bearing capacity factors: from table (1), for φ = 30°, Nq = 22.46,..Nγ = 19.13
qu=𝒄′Nc𝑺𝒄+𝒒′Nq𝑺𝒒+ 0.5B𝜸′Nγ𝑺𝜸=0+1x1.0 (19-10)x22.46+0.5x1(19-10)19.13x1.0 = 288.225 KN/𝒎2
Solution:-
𝒒𝒂𝒍𝒍 =
qult
𝑭.𝑺
=
288.225
3
= 96.075 KN/𝒎2
Example (2):
qu=𝒄′Nc+𝒒′Nq+0.5B𝜸′Nγ
Solution:-
14. Determine the size of square footing to carry net allowable load of 295 KN. FS=3. Use Terzaghi
equation assuming general shear failure.
Example (4):
18. 2. Skempton’s Bearing Capacity:
For saturated clay soils(φ = 0) , Skempton (1951) proposed the following equation
for a strip foundation
qu=𝒄𝒖Nc+𝒒
Fig. Skempton's bearing capacity factor Nc for clay soils
• Nc: Bearing capacity factor for
strip and square (or circular)
foundations as a function of the
D/B ratio are given in Fig. below.
• 𝒄𝒖: Undrain cohesion
• 𝒒 :Total stress (In all cases,
whether the soil is dry or
saturated) (q=𝛄𝐭𝐃𝐟).
𝑵𝑪 (𝑹𝒆𝒄𝒕𝒂𝒏𝒈𝒖𝒍𝒂𝒓) = 𝑶. 𝟖𝟒 + 𝟎. 𝟏𝟔
𝑩
𝑳
∗ 𝑵𝑪(𝒔𝒒𝒖𝒂𝒓𝒆)
• 𝑭𝒐𝒓 𝑹𝒆𝒄𝒕𝒂𝒏𝒈𝒖𝒍𝒂𝒓 𝑭𝒐𝒐𝒕𝒊𝒏𝒈
